Multiscale modelling and simulations of tissue perfusion using the Biot-Darcy-Brinkman model

摘要

We introduce a new mathematical model enabling to describe tissue perfusion whereby the microstructure is respected by virtue of effective model parameters. The proposed model entitled Biot-Darcy-multiBrinkman (BDmB) is an extension of the recently derived model (Rohan et al., 2019). The BDmB model enables to describe parallel flows in several mutually disconnected mesoscopic channels immersed in the mesoscopic poroelastic matrix where the flow is governed by the Darcy model. It is derived by the two-level homogenization of the microscopic fluid-structure interaction problem to respect the hierarchical structure of the two-phase medium. The second-level upscaling is explained, providing the macroscopic homogenized coefficients and the model equations. Boundary conditions are discussed. The model has been implemented in the finite element code SfePy which is well-suited for computational homogenization and solving coupled problems. It describes flows and deformations in the liver tissue with nearly periodic mesoscopic structure. Two mesoscopic channels are considered to represent the portal and hepatic vasculatures, while the matrix represents the sinusoidal porosity. Numerical simulations of stationary and nonstationary problems are reported with applications to the liver perfusion. Using numerical tests, the homogenized BDmB model is validated against direct simulations of heterogeneous mesoscopic structures. (c) 2020 Elsevier Ltd. All rights reserved.